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The following passage has been extracted from the Newton's (John Stewart's English translated version) "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms":

I consider mathematical quantities in this place not as consisting of parts; but as described by a continued motion. Lines are described, and there by generated not by the apposition of parts, but by the continued motion of points; superficies's by the motion of lines; Solids by the motion of superfices's; Angles by the rotation of the sides; Portion of time by a continual flux: and so in other quantities. These geneses really take place in nature of things, and are daily seen in the motion of bodies. And after this manner the ancients, by drawing moveable right lines along immoveable right lines taught the genesis of reflection...

Here Newton doesn't provide any reason on why he wants to describe lines to be generated by the "continued" motion rather than by the appositon of parts (= points??). Is there any reason for his preference for motion view?

And I noticed that Newton doesn't define point. I don't understand whether he is following Euclid's method of having some of the terms to be undefined, or some other philosophy. I want to know Newton's view on mysterious points. I will be really happy if sources on this regard (Newton's view on points) is provided.


Meaning of Apposition from "The New Oxford American Dictionary": The positioning of things or the condition of being side by side or close together. So, I interpret apposition of parts to be positioning of points/parts side by side or close together to form a line.


References to the full latin text:

I have asked same question in HSM.

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  • $\begingroup$ I have already asked this question in Math stack exhange. I asked this once again in hoping to get answer, which I didn't get as expected from MSE. I feel no hesitation to ask it here, as Newton being physicist and mathematician, employed calculus (and the present question is on its basics) to build physics. $\endgroup$ – Immortal Player Aug 20 '15 at 14:23
  • $\begingroup$ How do you interpret "apposition of parts"? Does it mean he wants to think of motion as continuous rather than in discrete steps - an obvious prerequisite for calculus? $\endgroup$ – Floris Aug 20 '15 at 14:43
  • $\begingroup$ @Floris: Meaning of Apposition from "The New Oxford American Dictionary": The positioning of things or the condition of being side by side or close together. So, I interpret apposition of parts to be positioning of points/parts side by side or close together to form a line. $\endgroup$ – Immortal Player Aug 20 '15 at 14:49
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    $\begingroup$ Probably only Newton himself knows why. $\endgroup$ – Shing Aug 20 '15 at 14:53
  • $\begingroup$ I do not agree with your interpretation that points = parts in Newton view. How do you infer it? $\endgroup$ – arivero Aug 20 '15 at 17:33
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To avoid to decide if his derivative $\dot u(x)$ is a covariant or a contravariant object (or perhaps to go for the contravariant one). Seriously.

Of course not rigorously, nor even formally. Duality will enter scene in the XIXth-XXth centuries. We got used to integrate a density across a path, or to multiply vector and covectors from the tangent and cotangent bundle. But some precursor of duality resounds already from the greeks and the integration of the cone, done by Democritus. The points and parts of a line are in a mutual relationship: points are separated because there is part between then, and a finite part can always be separated into two smaller parts by inserting a point.

In the age previous to Newton, a lot of people tried to find some foundations for calculus building over the ancient geometry. The most famous of the approaches, that of Cavalieri, directly paid hommage to Democritus by calling its approach "method of indivisibles". Even the word of "apposition" brings some reminiscence of the concept of contact between atoms, reviewed by Aristotle. It makes sense that Newton, aiming to invent instead of rediscover, tried to bypass the question of the elementary constituents of the line.

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